Chapter 10
Application: Levinson’s theorem
In this chapter, we briefly describe how the formalism introduced in the previous chapters leads to some index theorems in the context of scattering theory. Obviously, we
shall only scratch the surface, and most the previous material is not really necessary
for the example presented hereafter. However, for more involved examples this material
turns out to be essential. We refer to [Ric15] for more information on the subject.

10.1

The -anisotropic algebra

In this section we briefly construct a C ∗ -algebra which will play a major role later on.
This algebra has been introduced in [GI03, Sec. 3.5] for a diﬀerent purpose, and werefer to this paper for the details of the construction.
In the Hilbert space L2 (R) we consider the two canonical self-adjoint operators X ofdmultiplication by the variable, and D = −i dxof diﬀerentiation. These operators satisfy
the canonical commutation relation written formally [iD, X] = 1, or more precisely
e−isX e−itD = e−ist e−itD e−isX . We recall that the spectrum of both operators is R. Then,for any functions φ, η ∈ L∞ (R),
( 2one )can consider by bounded functional calculus theoperators φ(X) and η(D) in B L (R) . And by mixing some operators φi (X) and ηi (D)for suitable functions φi and ηi , we are going to produce an algebra C which will be
useful in many applications.
()
Let us consider the closure in B L2 (R) of the C ∗ -algebra generated by elements ofthe form φi (D)ηi (X), where φi , ηi are continuous functions on R which have limits at±∞. Stated diﬀerently, φi , ηi belong to C([−∞, +∞]). Note that this algebra is clearly
unital. In the sequel, we shall use the following notation:()∗
C(D,X) := C φi (D)ηi (X) | φi , ηi ∈ C([−∞, +∞]) .
Let us also consider the C ∗ -algebra generated by φi (D)ηi (X) with φi , ηi ∈ C0 (R), whichmeans that these functions are continuous and vanish at ±∞. As easily
observed,
this()
algebra is a closed ideal in C(D,X) and is equal to the C ∗ -algebra K L2 (R) of compact
operators in L2 (R), see for example [GI03, Corol. 2.18].
101

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CHAPTER 10. APPLICATION: LEVINSON’S THEOREM()
Let us now study the quotient C ∗ -algebra C(D,X) /K L2 (R) . For that purpose, we
consider the square  := [−∞, +∞]×[−∞, +∞] whose boundary  is the union of fourparts:  = C1 ∪ C2 ∪ C3 ∪ C4 , with C1 = {−∞} × [−∞, +∞], C2 = [−∞, +∞] × {+∞},
C3 = {+∞} × [−∞, +∞] and C4 = [−∞, +∞] × {−∞}. We can also view C() as thesubalgebra ofC([−∞, +∞]) ⊕ C([−∞, +∞]) ⊕ C([−∞, +∞]) ⊕ C([−∞, +∞])

(10.1)

given by elements Γ := (Γ1 , Γ2 , Γ3 , Γ4 ) which coincide at the corresponding end points,
that is, Γ1 (+∞) = Γ2 (−∞),= Γ3 (+∞), Γ3 (−∞) = Γ4 (+∞), and Γ4 (−∞) =( 2 Γ2 (+∞))
Γ1 (−∞). Then C(D,X) /K L (R) is isomorphic to C(), and if we denote the quotient
map by()
q : C(D,X) → C(D,X) /K L2 (R) ∼= C()()
then the image q φ(D)η(X) in (10.1) is given by Γ1 = φ(−∞)η(·), Γ2 = φ(·)η(+∞),
Γ3 = φ(+∞)η(·) and Γ4 = φ(·)η(−∞). Note that this isomorphism is proved in [GI03,
Thm. 3.22]. In summary, we have obtained the short exact sequence()
q
0 → K L2 (R) ,→ C(D,X) → C() → 0(10.2)()( 2)with K L2 (R) and C(D,X)
represented
in
B
L
(R)
, but with C() which is not natu( 2)rally represented in B L (R) . Note however that each of the four( functions
summing)2
up in an element of C() can individually be represented in B L (R) , either as amultiplication operator or as a convolution operator.
We shall now construct several isomorphic versions of these algebras. First of all,let us consider the Hilbert space L2 (R+ ) and the action of the dilation group. Moreprecisely, we consider the unitary group {Ut }t∈R acting on any f ∈ L2 (R+ ) as
( )[Ut f ](x) = et/2 f et x ,∀x ∈ R+(10.3)

which is usually called the unitary group of dilations, and denote its self-adjoint generator by A and call it the generator of dilations.
Let also B be the operator of multiplication in L2 (R+ ) by the function − ln,i.e. [Bf ](λ) = − ln(λ)f (λ) for any f ∈ Cc (R+ ) and λ ∈ R+ . Note that if one sets
L for the self-adjoint operator of multiplication by the variable in L2 (R+ ), i.e.
[Lf ](λ) := λf (λ)

f ∈ Cc (R+ ) and λ ∈ R+ ,(10.4)

then one has B = − ln(L). Now, the equality [iB, A] = 1 holds (once suitably defined),and the relation between the pair of operators (D, X) in L2 (R) and the pair (B, A)
in L2 (R+ ) is well-known and corresponds to the Mellin transform. Indeed, let V :L2 (R+ ) → L2 (R) be defined by (V f )(x) := ex/2 f (ex ) for x ∈ R, and remark that V
is a unitary map with adjoint V ∗ given by (V ∗ g)(λ) = λ−1/2 g(ln λ) for λ ∈ R+ . Then,the Mellin transform M : L2 (R+ ) → L2 (R) is defined by M := FV with F the usual

10.1. THE -ANISOTROPIC ALGEBRA

103

unitary Fourier transform1 in L2 (R). The main property of M is that it diagonalizes
the generator of dilations, namely, M AM ∗ = X. Note that one also has M BM ∗ = D.
Before introducing a first isomorphic algebra, observe that if η ∈ C([−∞, +∞]),then()
M ∗ η(D)M = η(M ∗ DM ) = η(B) = η − ln(L) ≡ ψ(L)for some ψ (∈ C([0, +∞]).
Thus, by taking these equalities into account, it is natural to)2
define in B L (R+ ) the C ∗ -algebra()
C(L,A) := C ∗ ψi (L)ηi (A) | ψi ∈ C([0, +∞]) and ηi ∈ C([−∞, +∞]) ,()
and clearly this algebra is isomorphic to the C ∗ -algebra C(D,X) in B L2 (R) . Thus,
through this isomorphism one gets again a short exact sequence()
q
0 → K L2 (R+ ) ,→ C(L,A) → C() → 0
with the square  made of the four parts  = B1 ∪ B2 ∪ B3 ∪ B4 with B1 = {0} ×[−∞, +∞], B2 = [0, +∞] × {+∞}, B3 = {+∞} × [−∞, +∞], and B4 = [0, +∞] ×{−∞}. In addition, the algebra C() of continuous functions on  can be viewed as a
subalgebra of
(
)
(
)
(
)
(
)
C [−∞, +∞] ⊕ C [0, +∞] ⊕ C [−∞, +∞] ⊕ C [0, +∞](10.5)
given by elements Γ := (Γ1 , Γ2 , Γ3 , Γ4 ) which coincide at the corresponding end points,
that is, Γ1 (+∞) = Γ2 (0), Γ2 (+∞) = Γ3 (+∞), Γ3 (−∞) = Γ4 (+∞), and Γ4 (0) =Γ1 (−∞).
Finally, if one sets Fs for the unitary Fourier sine transformation in L2 (R+ ), definedfor x, k ∈ R+ and any f ∈ Cc (R+ ) ⊂ L2 (R+ ) by
∫ ∞1/2
[Fs f ](k) := (2/π)
sin(kx)f (x) dx(10.6)
0√then the equalities −A = Fs∗ AFs and HD = Fs∗ LFs hold, where HD corresponds to
the Dirichlet Laplacian on R+ (see the next section for its definition). As a consequence,by)note that the formal equality [i 12 ln(HD ), A] = 1 can also be fully justified. Moreover,( 2∗
using this new unitary transformation one gets that the C -subalgebra of B L (R+ )defined by()
C(HD ,A) := C ∗ ψi (HD )ηi (A) | ψi ∈ C([0, +∞]) and φi ∈ C([−∞, +∞]) ,(10.7)
is again isomorphic to C(D,X) . In addition, the following short exact sequence takes place()
q
0 → K L2 (R+ ) ,→ C(HD ,A) → C() → 0,
(10.8)and C() can naturally be viewed as a subalgebra of the algebra introduced in (10.5)
with suitable compatibility conditions at end points.
1∫
For f ∈ Cc (R) and x ∈ R we set [Ff ](x) = (2π)−1/2 R e−ixy f (y) dy.

104

10.2

CHAPTER 10. APPLICATION: LEVINSON’S THEOREM

Elementary scattering system

In this section we introduce an example of a scattering system for which everything canbe computed explicitly. It will allow us to describe more precisely the kind of results weare looking for, without having to introduce too much information on scattering theory.
In fact, we shall keep the content of this section as simple as possible.
Let us start by considering the Hilbert space L2 (R+ ) and the Dirichlet Laplaciand2
HD on R+ := (0, ∞). More precisely, we set HD = − dx
2 with the domain Dom(HD ) ={f ∈ H2 (R+ ) | f (0) = 0}. Here H2 (R+ ) means the usual Sobolev space on R+ of orderd2
2. For any α ∈ R, let us also consider the operator H α defined by H α = − dx
2 withDom(H α ) = {f ∈ H2 (R+ ) | f ′ (0) = αf (0)}. It is well-known that if α < 0 the operator
H α possesses only one eigenvalue, namely −α2 , and the corresponding eigenspace isgenerated by the function x 7→ eαx . On the other hand, for α ≥ 0 the operators H αhave no eigenvalue, and so does HD .
A common object of scattering theory is defined by the following formula:
W±α := s − lim eitH e−itHD ,αt→±∞and these limits in the strong sense are known to exist for this model. These operators
are called the wave operators, they are isometries, and their existence allows one to study
the operator H α with respect to HD . Moreover, we shall provide below a very explicit
formula for these operators. Let us still stress that scattering theory is a comparison
theory, one always study pairs of operators.
Our first result for this model then reads, see [Ric15, Cor. 9.3] for its proof:
Lemma 10.2.1. The following equalities hold:√])[ α + i HD√
1 + tanh(πA) − i cosh(πA)−1 ,α − i HD][ α − i√H
(
)
D
α
−1
1
√
W+ = 1 + 2 1 − tanh(πA) + i cosh(πA)−1 .
α + i HDW−α = 1 + 12(

−1

It clearly follows from these explicit formulas that the operators W±α belong to
the algebra C(HD ,A) introduced in (10.7). Since these operators are also isometries witha finite dimensional co-kernel,( they) can be considered as lifts for their image in thequotient algebra C(HD ,A) /K L2 (R+ ) . We shall come back to this approach involving
algebras in the next session, and work very explicitly for the time being.
Motivated by the above formula for W−α , let us now introduce the complex function√]()[ α + i xα−11√ −1 .
Γ : [0, +∞] × [−∞, +∞] ∋ (x, y) 7→ 1 + 2 1 + tanh(πy) − i cosh(πy)α−i xSince this function is continuous on the square  := [0, +∞]×[−∞, +∞], its restriction
on the boundary  of the square is also well defined and continuous. Note that thisboundary is made of four parts:  = B1 ∪ B2 ∪ B3 ∪ B4 with B1 = {0} × [−∞, +∞],

10.2. ELEMENTARY SCATTERING SYSTEM

105

B2 = [0, +∞] × {+∞}, B3 = {+∞} × [−∞, +∞], and B4 = [0, +∞] × {−∞}, and thatthe algebra C() of continuous functions on  can be viewed as a subalgebra of (10.5)
with the necessary compatibility conditions at the end points. With these notations,the restriction function Γα := Γα  is given for α ̸= 0 by
( α + i√·)α√ , − tanh(π·) + i cosh(π·)−1 , 1(10.9)
Γ = 1,α−i ·and for α = 0 by()Γ0 := − tanh(π·) + i cosh(π·)−1 , −1, − tanh(π·) + i cosh(π·)−1 , 1 .

(10.10)For simplicity, we have directly written this function in the representation provided by(10.5).
Let us now observe that the boundary  of  is homeomorphic to the circle S.
Observe in addition that the function Γα takes its values in the unit circle T of C.
Then, since Γα is a continuous function on the closed curve  and takes values in T,
its winding number Wind(Γα ) is well defined and can easily be computed. So, let us
compute separately the contribution wj (Γα ) to this winding number on each component
Bj of . By convention, we shall turn around  clockwise, starting from the left-down
corner, and the increase in the winding number is also counted clockwise. Let us stressthat the contribution on B3 has to be computed from +∞ to −∞, and the contribution
on B4 from +∞ to 0. Without diﬃculty one gets:
α<0
α=0
α>0

w1 (Γα )0−1/2
0w2 (Γα ) w3 (Γα ) w4 (Γα ) Wind(Γα )
1/2
1/2
0
1
0
1/2
0
0
−1/2
1/2
0
0

By comparing the last column of this table with the information on the eigenvaluesof H α mentioned at the beginning of the section one gets:
Proposition 10.2.2. For any α ∈ R the following equality holds:
Wind(Γα ) = number of eigenvalues of H α .

(10.11)

The content of this proposition is an example of Levinson’s theorem. Indeed, it
relates the number of bound states of the operator H α to a quantity computed on the
scattering part of the system. Let us already mention that the contribution w2 (Γα ) is the
only one usually considered in the literature. However, we can immediately observe that
if w1 (Γα ) and w3 (Γα ) are disregarded, then no meaningful statement can be obtained.
Obviously, the above result should now be recast in a more general framework, andthe algebraic background should be taken into account. Indeed, except for very specificmodels, it is usually not possible to compute precisely both sides of (10.11), but suchan equality still holds in a much more general setting. The next section shows how
K-theory can provide an insight on Levinson’s theorem.

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CHAPTER 10. APPLICATION: LEVINSON’S THEOREM

10.3

The abstract topological Levinson’s theorem

Before stating the main result of this chapter, let us reformulate the content of Proposition 6.2.4.(ii). The key point in the next statement is that the central role is playedby the partial isometry in C instead of the unitary element in Q. In fact, the following
statement is at the root of our topological approach of Levinson’s theorem.
Proposition 10.3.1. Consider the short exact sequenceq

0 → J ,→ C → Q → 0
with C unital. Let W be a partial isometry in Mn (C) and assume that Γ := q(W ) is a
unitary element of Mn (Q). Then 1n − W ∗ W and 1n − W W ∗ are projections in Mn (J ),andind([q(W )]1 ) := δ1 ([q(W )]1 ) = [1n − W ∗ W ]0 − [1n − W W ∗ ]0 .
In order to go one more step in our construction, let us add some information aboutsome special K-groups, as already mentioned in Example 4.3.6 and in Example 7.4.3.
Example 10.3.2. (i) Let C(S) denote the C ∗ -algebra of continuous functions{ on∞
the unit circle S, with the} L -norm, and let us identify this algebra with ζ ∈C([0, 2π]) | ζ(0) = ζ(2π) , also endowed with the L∞ -norm. Some unitary elements of C(S) are provided for any m ∈ Z by the functions
ζm : [0, 2π] ∋ θ 7→ e−imθ ∈ T.
Clearly, for two diﬀerent values of( m the) functions ζm are not homotopic, andthus define diﬀerent classes in K1 C(S) . With some( more) eﬀorts one can showthat these elements define in fact all elements of K1 C(S) , and indeed one has()
K1 C(S) ∼= Z.
Note that this isomorphism is implemented by the winding number Wind(·), which
is roughly defined for any continuous function on S with values in T as the numberof times this function turns around 0 along the path from 0 to 2π. Clearly, for anym ∈ Z one has Wind(ζm ) = m. More generally, if det denotes the determinant()
on Mn (C) then the mentioned isomorphism is given by Wind ◦ det on Un C(S) .
(ii) Let K(H) denote the C ∗ -algebra of all compact operators on a infinite dimensional and separable Hilbert space H. For any n one can consider the orthogonal
projections on subspaces of dimension n of H, and these finite dimensional projections belong to K(H). It is then not too diﬃcult to show that two projections of
the same dimension are Murray-von Neumann equivalent, while projections corresponding to two diﬀerent values of n are not. With some more eﬀorts, one showsthat( the dimensionof these projections plays the crucial role for the definition of)
K0 K(H) , and one has again()
K0 K(H) ∼= Z.

10.3. THE ABSTRACT TOPOLOGICAL LEVINSON’S THEOREM

107

In this case, the isomorphism is provided by the usual trace Tr on finite dimensional projections, and by the tensor product( of this) trace with the trace tr on
Mn (C). More precisely, on any element of Pn K(H) the mentioned isomorphism
is provided by Tr ◦ tr.
Let us now add the diﬀerent pieces of information we have presented so far, and get
an abstract version of our Levinson’s theorem. For that purpose, we consider an arbitrary separable Hilbert space H and a unital C ∗ -subalgebra C of B(H) which contains
the ideal of K(H) of compact operators. We can thus look at the short exact sequenceof C ∗ -algebrasq
0 → K(H) ,→ C → C/K(H) → 0.
Let us assume in addition that C/K(H) is isomorphic to C(S). Then, if we take theresults presented in the previous example into account, one infers that
(
) ind
(
)
Z∼
= K1 C(S) −→ K0 K(H) ∼=Z
with the first isomorphism realized by the winding number and the second isomorphism
realized by the trace. As a consequence, one infers from this together with Proposition
10.3.1 that there exists n ∈ Z such that for any partial isometry W ∈ C with unitaryΓ := q(W ) ∈ C(S) the following equality holds:()
Wind(Γ) = nTr [1 − W ∗ W ] − [1 − W W ∗ ] .
(10.12)
We emphasize once again that the interest in this equality is that the left hand side isindependent of the choice of any special representative in [Γ]1 . On the other hand, inthe context of scattering theory the r.h.s. of (10.12) is well understood, see the next
statement. Let us also mention that the number n depends on the choice of the extensionof K(H) by C(S), see [W-O93, Chap. 3.2], but also on the convention chosen for thecomputation of the winding number.
If we summarize all this in a single statement, one gets:
Theorem 10.3.3 (Abstract topological Levinson’s theorem). Let H be a separable
Hilbert space, and let C ⊂ B(H) be a unital C ∗ -algebra such that K(H) ⊂ C andC/K(H) ∼= C(S) (with quotient morphism denoted by q). Then there exists n ∈ Zsuch that for any partial isometry W ∈ C with unitary Γ := q(W ) ∈ C(S) the following
equality holds:()
Wind(Γ) = nTr [1 − W ∗ W ] − [1 − W W ∗ ] .
(10.13)
In particular if W = W− = s − limt→−∞ eitH e−itH0 for some suitable scattering pair
(H, H0 ), then the previous equality reads()()
Wind q(W− ) = −n number of eigenvalues of H − number of eigenvalues of H0 .
Note that in applications, the factor n is determined by computing both sides ofthe equality on an explicit example.

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CHAPTER 10. APPLICATION: LEVINSON’S THEOREM

Let us finally show that the example presented in Section 10.2 can be recast in theprevious framework. We consider the Hilbert space L2 (R+ ) and the unital C ∗ -algebra
C(HD ,A) introduced in (10.7). As already mentioned, the( wave operator
W−α is an isometry)
which clearly belongs to the C ∗ -algebra (C(HD ,A) )⊂ B L2 (R+ ) . In addition, the imageof W−α in the quotient algebra C(HD ,A) /K L2 (R+ ) ∼= C() is precisely the function Γα ,defined in (10.9) for α ̸= 0 and in (10.10) for α = 0, which are unitary elements of C().
Finally, since C() and C(S) are clearly isomorphic, the winding number Wind(Γα ) ofΓα can be computed, and in fact this has been performed and recorded in the table ofSection 10.2.
If one sets Ep (H α ) for the orthogonal projection on the subspace generated by thebound states of the operator H α , then one has
{
(
)
(
)
−1 if α < 0 ,
α ∗
α
α
α ∗
α
Tr [1 − (W− ) W− ] − [1 − W− (W− ) ] = −Tr Ep (H ) =(10.14)0
if α ≥ 0 .
Thus, this example fits in the framework of Theorem 10.3.3, and in addition both sides of(10.13) have been computed explicitly. By comparing (10.14) with the results obtainedfor Wind(Γα ), one gets that the factor n mentioned in (10.13) is equal to −1 for these
algebras. Finally, since Ep (H α ) is related to the point spectrum of H α , the content of
Proposition 10.2.2 can be rewritten as
Wind(Γα ) = ♯σp (H α ).
This equality corresponds to a topological version of Levinson’s theorem for the elementary model. Obviously, this result was already obtained in Section 10.2 and all theabove framework was not necessary for its derivation. However, we have now in ourhands a very robust framework which can be applied to several other situations, see[Ric15] and the references therein.
Remark 10.3.4. As a concluding remark, let us mention how the algebraic framework
could still be extended. For that purpose, consider a short exact sequence
0 −→ J −→ C −→ Q −→ 0
and the corresponding index map ind : K1 (Q) → K0 (J ). Assume that η is an even ntrace on J which can be paired with K0 (J ), see Theorem 9.5.8. Then one can wonder if
there exists a map on higher traces which is dual to the index map, i.e. a map # whichassigns to an even trace η an odd trace #η such that the equality⟨⟩ ⟨⟩[ind(Γ)]0 , [η] = [Γ]1 , [#η]
(10.15)e ? Except for some special cases (like in Theorem 10.3.3 for aholds, for any Γ ∈ Un (Q)0-trace and a 1-trace), the answer to this question is apparently not known.